Meaning of Poem

The poem is addressed to the Blessed Virgin, and its title evokes the prayer called the Litany of Loreto. In this prayer, the Virgin Mary's prayers are requested, and she is invoked under many metaphorical titles. The poem mentions, or alludes to, some of these. For clarity I will set these titles of Mary in bold.

The first stanza pictures God, born and "become little", coming into the world through Mary, who in this sense is the Gate of Heaven. The period (full stop) at the end has in some ways the force of a comma, leading into the next stanza.

You're having difficulty with the final two lines of this stanza:

Or shutting out his shining skies awhile
Built you about him for a house of gold
To see in pictured walls his storied world
Return upon him as a tale is told.

"Pictured walls" here means simply "walls with pictures on them", and "storied world" could mean "world filled with stories" or "world told about in stories". The image is of Christ coming from the "shining skies" of Heaven to begin life within a House of Gold, and then to see the universe he created, not as the Creator might see it, but as one might encounter it for the first time in picture and in story, in a tale told him.

The next stanza imagines Mary as the Mirror of Justice, and pictures the Christchild looking up at her, the one being who is able to reflect His beauty and love as fully as humanly possible. It pictures God himself, looking on himself reflected in Mary.

The fourth stanza pictures Mary's title Morning Star. Christ, of course, is the Morning Star preeminent; but Chesterton pictures Mary as a star in a sort of "inverted reality" moment where Heaven is looking up to Earth, the Creator looking up to the created.

The fifth stanza you also had difficulty with.

Or young on your strong knees and lifted up
Wisdom cried out, whose voice is in the street,
And more than twilight of twiformed cherubim
Made of his throne indeed a mercy-seat.

First, the sentence structure is inverted. "Wisdom" is the subject, that is, Christ, the Word or the Logos. He is "lifted up" to sit on the "strong knees" of Mary, the Seat of Wisdom. "Whose voice is in the street" is a reference to, perhaps among others, Proverbs 1:20: "Wisdom cries aloud in the street, in the open squares she raises her voice" (New American Bible, Revised Edition).

In the same way that Christ sits on Mary, so God was pictured sitting atop the Ark of the Covenant, the cover of which was formed in the shape of two cherubim:

You shall then make a cover of pure gold, two and a half cubits long, and one and a half cubits wide. Make two cherubim of beaten gold for the two ends of the cover; make one cherub at one end, and the other at the other end, of one piece with the cover, at each end. The cherubim shall have their wings spread out above, sheltering the cover with them; they shall face each other, with their faces looking toward the cover. This cover you shall then place on top of the ark. In the ark itself you are to put the covenant which I will give you. There I will meet you and there, from above the cover, between the two cherubim on the ark of the covenant, I will tell you all that I command you regarding the Israelites. (Exodus 25:17–22 New American Bible, Revised Edition)

This cover, regarded as the seat of God, was often called the "mercy seat"; Chesterton points out how much more appropriate the name is for the lap on which Jesus sat.

I'm not sure why Chesterton uses "twilight" here—perhaps because the Ark was never available for public viewing?—but "twiformed", which means simply "formed of two bodies or shapes", is obviously intended to echo it.

The final stanza brushes past two more titles of Mary, when it pictures Jesus rising from a child's play to show his love for his mother. In my opinion the initial "Or" of the third line would make more sense if it were "Of", but I don't have a copy of the poem as it was first published.

Here, Jesus is pictured pulling himself up to climb up his mother as one would climb a Tower of Ivory, and kissing her lips. Lips have often been compared to roses in romantic poetry, but these lips are of the Mystical Rose herself.